Implementing Mathematics Reform: The Challenges Within

Recommendations for reform in mathematics education uniformly call for
an increased emphasis on meaningful experiences in school mathematics and
decreased emphasis on repeated practice of computational algorithms (NCTM
1989). The reform movement also calls for teaching approaches that support
conceptual understanding of mathematical ideas (NCTM 1991).

In recent years, curriculum programs that support the visions of the
reform for school mathematics have been developed. These programs, commonly
referred to as Standards-based curricula, are designed so to foster a conceptual
approach to teaching and learning mathematics. A strong component of the
new curricula is their emphasis on student centered learning and students'
independent investigations of mathematical ideas. Inherent in the design
of these programs is the view taken of the teacher as the facilitator of
learning within the classroom. As students engage in investigations, the
teacher is expected to create an environment in which mathematical discourse
takes place. She guides the learners' thinking by creating an open forum
for the exchange of ideas. The teacher also helps students synthesize their
findings and connect those findings to a coherent mathematical structure
as she devises strategies for evolving students' thinking from an intuitive
to a more rigorous level. In many ways, these new visions of teaching place
greater demands on teachers than of the traditional method of instruction
in which the teacher disseminated bits and pieces of knowledge among students.
In light of these new teacher roles and responsibilities, the natural questions
that emerge include what motivates a teacher to support the use of the Standards-based
curricula? What allows a teacher to be successful in implementing such curricula?
What factors tend to encourage or discourage teachers to conform to the
new visions of teaching and curriculum?

Gaining a deeper understanding of these questions was the underlying
motive of the current study. This research is a case study of two 7th
grade mathematics teachers as they engaged in implementation of a Standards-based
curriculum program. Of particular interest was investigating those characteristics
that seemingly facilitated the work of teachers as they enacted the new
curriculum in their classes, along with those that impeded their work. This
study draws on two years of interviews and observational data, focuses on
a complex factor-- the teacher's interactions with their curriculum, and
their thinking about their practice.

The setting

During the school years, 1995-1996, a group of 158 middle school teachers
and some administrators representing 24 Missouri school districts, participated
in a curriculum review project (Missouri Middle School Mathematics Project--)
funded by the National Science Foundation. This project involved the participating
teachers in the review and evaluation of four standards-based middle levels
(6-8) curricular materials in schools across the state of Missouri. The
project provided teachers with opportunities to gain knowledge about, and
experience with curricular materials as they implemented several units from
each program over a period of one school year (Reys et al. 1997).

Participating teachers received a two day inservice training on each
of the programs prior to using the materials in their classrooms. Student
and teacher editions were also provided for the teachers. Teachers shared
their experiences and insights about the programs with other teachers through
project-sponsored conferences which occurred quarterly and concurrent to
curriculum workshops. In addition, local regional groups were formed to
further assist teachers during the review and implementation of the curriculum
materials. The purpose of the regional meetings was to provide teachers
with a local professional network through which ideas were exchanged and
suggestions for better implementation of the materials were shared. Each
regional cohort met individually and shared insights on the quality and
substance of the units they taught and student outcomes. The activities
of each region were coordinated by a regional leader. This regional leader
served as a liaison between the project directors and the participating
teachers at local school districts.

The researchers were two of the six regional leaders in the project and
were involved in all stages of planning and implementation of the activities
for teachers. This involvement allowed us to gain an insight on the level
of implementation of the programs by teachers at various school districts.

As teachers began using the programs in their classrooms, several patterns
of actions emerged. Among the 66 teachers, 10 showed great commitment to
the use of the materials, consistently implemented the programs in their
classrooms and systematically evaluated the impact of the use of the materials
on their students' learning. There were 15 teachers whose use of the programs
were episodic. These teachers reported on using the programs occasionally
and only when their curriculum allowed room for extra activities. 21 teachers
began the use of the programs enthusiastically, however their own reports
on the frequency of the use of the materials was indicative of a decline
in their use.

This diverse data in the frequency of the use of the programs by teachers
was interesting to us. Although our initial research on the factors that
enhanced or impeded the work of teachers as they used the programs in their
classrooms revealed that the presence of supportive structures within schools
was a critical factor in whether the programs were used by them or not (Manouchehri,
in press, Manouchehri & Goodman, in press, Manouchehri & Sipes,
in press), the fact that even under similar work conditions some teachers
seemed reluctant to use the programs originated curiosity. This motivated
us to look even more closely at the teaching/ learning processes taking
place in the classrooms. Due to the fact that after the first year of pilot
some school districts had opted to adopt one of the four programs as their
textbooks, we were granted the opportunity to develop a systematic plan
to study the issue. Our goal was to investigate the elements that influenced
those teachers who showed great enthusiasm towards the use of the new curricula
throughout the entire year, and those whose enthusiasm declined over time.
In light of these goals, two specific research questions guided our research.
These included:

1. What factors tend to motivate or discourage the use of the programs
by teachers?

2. How do teachers judge the new curriculum and its value for their
practice?

Due to the large number of teachers in each category we decided to choose
one representative from each of the two groups and study them over time.
For this purpose we selected 2 teachers, Gina and Bonnie, who taught the
same grade level mathematics in the same school. Our purpose was to study
each teacher as they made curricular and instructional decisions and as
they taught the materials.

The Teachers: Gina and Bonnie

Gina, and Bonnie were two of the teachers that joined the project in
the Summer of 1995. Both teaches had similar educational backgrounds and
had taught middle school mathematics for 6 years. Both teachers worked within
a school district that supported the efforts of the teachers in implementing
the recommendations of the reform. Moreover, at the school district where
they worked a strong mechanism for supporting teachers in their efforts
was created. This included ongoing professional development plans designed
specifically for mathematics teachers, and providing teachers with team
planning time during which groups of teachers engaged in collaborative planning
of their instruction.

Our initial data on both teachers came from the regional meetings during
the first year which all teacher participants in the project engaged in
discussions about their experiences as they implemented the programs. These
discussions consisted of their assessment of the four programs and what
they viewed as the impact of the new curricula on their students. During
this initial phase both Gina and Bonnie were highly proactive in expressing
their positive reactions to the curricula and encouraged other teachers
to use them in their classrooms. Both teachers, excited about the impact
of the programs on their students, constantly reassured other teachers of
the value of recommended changes for their practice. The following excerpts
from the first regional meeting (September 1995) manifest the enthusiasm
of each teacher towards the use of the programs at that point.

I really like the programs. They bring a quality in class that I
have not seen before. Kids talk and discuss things-- even my least able
kids are now doing mathematics. I am really excited about what is happening
in my class (Bonnie, September 1995).

The programs are really helping my students communicate mathematics.
They are talking to each other, they are making conjectures about what
is right and wrong, and about mathematical relationships. They are helping
those that were never successful in doing math to be engaged-- it makes
them feel good-- Those that used to be good mathematics students are still
doing well. They are problem solving and stuff (Gina, September 1995).

Both Gina and Bonnie reported on their commitment to the NCTM standards,
and of its value for mathematics education of learners. Indeed, both Bonnie
and Gina consistently served as pedagogical problem solvers (Cooney 1994)
among the group and offered practical solution methods for overcoming obstacles
that other teachers identified as impeding their work. These obstacles at
that point consisted primarily of parents' complains, and their district's
lack of support as they implemented mathematics reform. The following excerpts
from our second regional meeting during December 1995 highlight the active
role Bonnie and Gina played in helping others as they suggested strategies
for dealing with the dilemmas.

T1: The parents are the problem in our school. They are complaining
because we are not sending kids home with worksheets and specific exercises.

Bonnie: Well, it only tells us that we need to try to educate parents
too... (pause) I mean, if we are really serious about the reform we know
that we should work with parents at the same time.

Gina: Bonnie is right-- What we did in our district was that we organized
parent teacher nights where we actually did some of the activities we do
with kids with them. It really helped-- I think we still have some handouts
of the program we put together we can give you a copy if you want to use
it.

During our third regional meeting two months later (February 1996), as
some teachers raised concerns about the extensive use of students' self
assessment procedures and a lack of sufficient formal tests in the programs,
Bonnie and Gina voiced a highly student-centered approach to assessment.
Both teachers, operating more from a theoretical and philosophical perspective,
encouraged other teachers to trust students' judgments, and their ability
in providing teachers with accurate self assessment reports of their learning.
Both teachers made recommendations towards combining formal and informal
assessment techniques when evaluating students' learning.

I think kids are more critical in their assessment than we are. They
tend to give lower scores to themselves when I ask them to evaluate their
solution methods, or to report on their won level of understanding of different
concepts. I think we should place the responsibility on their shoulders
and help them become independent thinkers. I also use some of my own quizzes
and tests. It is time consuming at times because I don't have the chapter
tests like I did before but I know that I know better where my students
are now (Bonnie, February 1996)

Anytime I have asked students to give me feedback on their understanding
of topics I discuss in class, they have been pretty much on target. I agree
with Bonnie that they tend to me so much critical of each other than I
ever would be-- I think that is really important for them to learn to make
judgments like that-- After all, assessing their own learning is a real
good critical thinking, problem solving activity for them. I also think
that we need to use more formal tests and stuff. I don't see using the
tests I have made up for my students conflicting other kinds of assessing
I do in class. besides, I also want them to show mastery of certain skills
that are not present in the curricula. So, I make up some of my own (Gina,
February 1996)

In spite the fact that during the first 6 months of pilot both Bonnie
and Gina consistently supported the new curricula and reported on regular
use of them in their classrooms, this momentum was not sustained by Bonnie
towards the last three months of implementation. As Gina continued to report
enthusiastically on the use of the program in her class, the frequency of
the use of the program seemed to have declined in Bonnie's case. She was
not as positive towards the use of the new curricula, tended to remain silent
during the regional meetings, and seemed reluctant to engage in the discussions
that concerned the content of the programs and learners' mathematical thinking
and problem solving skills. Indeed, on occasions, Bonnie expressed comments
that were indicative of a lack of satisfaction with the impact of the use
of the programs on her students' learning.

You know I am not so sure if this is going to be useful to the kids
in the long run-- Some of them are questioning the value of what we do
in class. I am not sure if this is something that everyone is feeling or
it is only me-- I am beginning to wonder if this is doing what is supposed
to do -- I mean I am not sure if it is helping kids learn what they are
supposed to be learning at this point(Bonnie, April 1996-Field notes))

On the other hand, Gina persisted to advocate the use of the curricula
and encouraged others in maintaining commitment to the goals of the mathematics
reform. In particular, she constantly provided other teachers with testimonials
about her students' rapid development in the area of problem solving.

I just gave my students a problem the other day-- I used to give
it to the 8th graders when I taught eight grade but I just decided to test
it out with my group this year. You would not believe what they did with
the question. (Gina, April 1996-Field notes)

As Gina continued to use the programs, Bonnie reported on sporadic use
of them, often reverting to the old textbook for teaching of certain topics.
At this point the school district in which Bonnie and Gina worked made a
commitment to adopt one of the Standards-based curricula in the following
year. In light of this decision and the fact that both teachers were required
to teach one of already piloted programs we asked both Bonnie and Gina if
we could visit their classrooms occasionally as they taught the new textbook
in the following year. We began observing classes in September of 1996,
a month after the beginning of the school year, and continued our data collection
till March of 1997. Both teachers' classrooms were observed 8 times during
a 6 month period. We scheduled the observation so both teachers' classrooms
were visited as they taught the same activities. In each case, the teacher
was interviewed following her session to collect data on her perceptions
of how the lesson had progressed, as well as her plan for subsequent instruction.

Methodology and Theoretical Framework

The purpose of the current study was to investigate the nature of interactions
of two middle school teachers with, and their assessment of a Standards-based
mathematics curriculum. Of particular interest to us was determining the
basis for teachers' decisions concerning the use, or lack of use of the
program in their classrooms. For these reasons we needed to look at teachers'
practices over time as they taught the materials in their classrooms, and
engage in ongoing dialogues with them about their perceptions of the usefulness
of the programs for their instruction. These goals required us to adopt
an ethnographic research design (Atkinson & Hammersley 1994; Bodgan
& Bilken 1992 ; Geertz 1983; Goetz & LeCompte 1984). Our research
draws on conversations with teachers and classroom observations of their
practice. These methods were used for several reasons and stemmed from our
theoretical perspective.

Our work builds on a constructivist theoretical perspective (von Glasersfeld
1989; Noddings 1992) to investigate the relationship between teachers' thinking
and their teaching practices. From this perspective, we assume that teachers
assess the curriculum and the instructional roles placed on them based on
their existing knowledge about teaching, curriculum, learners, and their
experiences in the classroom. This knowledge is partly personal and idiosyncratic.
Certainly, teachers construct their practice and make pedagogical decisions
according to what makes sense to them (Briscoe 1993; Carlgren & Lindblad
1991; Hewson & Hewson 1989, Shymansky & Kyle 1992). Therefore, to
understand better the complex relationship between teachers' thinking, and
their interpretation of the curriculum program we needed to obtain their
own insights into their practices and the curriculum they taught. It was
imperative for us to gain the teachers' own perspectives on the issues concerning
their practice through conversations, dialogues, and interviews and as we
elicited information from them about the program and the impact of the use
of the program on their practice.

We also recognized that eliciting a teacher's thinking is not a simple
task. Some aspects of a teacher's thinking cannot be readily represented
by overt propositions made by teachers. These may include experiences, values,
and emotions, all of which can be both personal and professional (Carter
1993; Elbaz 1983; Tobin & Jakubowski 1992; Briscoe 1991; Johnston 1990).
What a teacher means by using certain language in discussing practice is
likely to be ambiguous to the researcher, and the language by itself cannot
represent the teacher's voice unless it is placed in the context of the
teacher's actions (Briscoe 1993; Munby 1986). Furthermore, since much of
what a teacher knows or believes may be tacit, not easily accessible to
the researcher and may be revealed only in teacher's actions (Marlans &
Osborne 1990; Pajares 1992; Ross, Cornett, & McCutcheon 1992; Peterson
1988; Peterson & Clark 1978), we decided to observe teachers' classrooms.

These assumptions motivated us to use two complementary methods of data
collection. We planned to examine teachers' practices through observations,
initiate conversations with them about the nature of their decisions, and
solicit specific information concerning their thinking as they engaged in
implementation of the Standards-based program in their instruction.

Data Collection and Data Analysis

In order to collect base line data on teachers' thinking about the standards-based
curricula and in particular about their goals for following year as they
were expected to teach the new textbook, we conducted an interview with
each one before the beginning of the new school year. Both teachers were
interviewed in July of 1996, prior to our classroom observations in order
to gain their perspectives on the events of the previous year along with
ways in which they perceived those experiences impacting their teaching
in the year to follow. Our intention was to solicit particular information
about their own perceptions about the Standards-based curricula, why they
used the programs, and what they liked and disliked about using them in
their instruction. These interviews helped us prepare a profile of those
elements that seemed to have greatly influenced the teachers' practices,
what they valued as worthwhile instructional activities, and ways in which
the use of the programs complemented or confronted their pedagogy. Each
interview lasted about two hours. Interviews were audio-taped and transcribed
later.

As we analyzed teachers' responses during the initial interview, we searched
for common themes, patterns, and frequently repeated expressions, that described
their expectation of curriculum and what they valued most (or disvalued)
about the Standards-based curricula in light of those expectations. We categorized
our data using key concepts and recurring themes (Glaser 1978; Glaser &
Strauss, 1967). The themes expressed by teachers during the initial interview
were used to guide our observations and as we examined and reconstructed
categories of teachers' actions later in the process. Moreover, we studied
how those themes came to play a part in their instruction, their interactions
with the new curriculum, and their judgment of the program.

Both Gina and Bonnie were observed 8 times during a 6 month time period.
On one occasion each teacher's class was visited on three consecutive days
in order to see how they developed a topic from the beginning to an end.
On two occasions, we asked teachers if we could observe the same lesson
taught to two different groups of students. This was to see if their presentation
of materials changed in light of their first hour experience with the lesson.

The appointments were scheduled so we could observe each teacher during
their first teaching period. This arrangement was manageable since their
first hour mathematics teaching sessions did not coincide. As Bonnie taught
Mathematics during the first hour, Gina taught Science and as Gina taught
her first math lesson, Bonnie taught social studies. This flexible schedule
allowed us to observe both teachers in the same day. Field notes were used
to record observations of teachers' behaviors and students' reactions to
them during the on-site visits.

Both teachers had the same planning hour. Our debriefing sessions occurred
frequently during the beginning part of their planning time or during lunch
as they were free to meet with us. If scheduling a meeting following an
observation was not possible, we communicated with teachers later either
on the phone, or over the Internet. However, this occurred only once throughout
the entire data collection phase. These debriefing sessions added greatly
to our understanding of teachers' thinking about their practice, increased
our knowledge about teachers' analysis of their own instruction and about
their particular strengths and weaknesses related to teaching Standards-based
curriculum.

Prior to observing classrooms we asked Bonnie and Gina if they wanted
us to collect specific information about their students or their instruction
during our visits. It was our intention to assure the teachers that our
purpose was to support them rather than judging their practice. In each
case, almost systematically, Bonnie asked for feedback on her interactions
with students. She desired to know whether she was sensitive to gender issues
(had called on equal number of boys and girls), or attended to the needs
of specific children. Gina solicited information on her efforts towards
asking high cognitive level questions during her instruction, motivating
students to participate in whole group discussions, and her method of responding
to students' questions. In particular, Gina wanted feedback on whether she
allowed students to clarify ambiguous points in the activities themselves
rather than her. As we collected the requested information, we shared the
data with the teachers. On several occasions Gina asked for specific teaching
suggestions, or elaboration on the mathematics involved in the program during
our follow up sessions. This included questions about whether there were
specific mathematical concepts that needed to be addressed as well as alternative
methods of presenting certain ideas to students. Bonnie, on the other hand,
was mostly concerned about the management issues in her class and asked
for ways in which she could involve all students in activities. She solicited
particular strategies that kept students "on the same page."

Teachers' Reflections: Initial Interview

During our initial interviews with teachers, we asked them to comment
on 5 specific questions. These included:

1. What they valued most about the Standards-based programs?

2. What they valued least about the programs?

3. What encouraged them to use the programs?

4. What discouraged them from using the programs?

5. What concerns they had for the following year?

Bonnie and Gina's responses to the interview questions revealed certain
categories of "pedagogical priorities" for each teacher. These
priorities were established based on their own images of "good teaching,"
and "productive mathematics learning" within their classrooms.
These images also enlivened the qualities that teachers attributed to good
curriculum, and consequently what they valued and/or disvalued about the
Standards-based curricula they had used in the previous year. It became
apparent later and as we observed teachers how intensely these images, expectation,
and priorities came to shape their instructional behaviors in class. These
images were revealed in what teachers described as important variables in
teaching, their expectations of an ideal curriculum, and important goals
for mathematics instruction.

Both teachers brought images to their teaching from their personal experiences
embedded in their vision of what teaching should be (Bullough 1994; Bullough
& Stokes 1994; Marland & Osborne 1990; Mostert 1992; Tobin 1990;
Provenzo et. al 1989). These images described their expectations of their
experiences as teachers and the instructional practices they chose to use
(Johnson, 1990; Calderhead & Robson 1991; Clandinin 1989; Elbaz 1983).
These images shaped their judgment of the Standards-based curriculum and
their assessment of their value for their instruction. Moreover, the discrepancies
between the images and realities of teaching in the presence of the curricula
seemed to have been a source of difficulty for one teacher, and compatibility
between the images and realities seemed to have served as a source of comfort
for another. These conflicting or confirming images tended to have been
the underlying motive for teachers' use of the curricula, or their lack
of use of them in their classrooms. Indeed, the same conflicting or confirming
images came to frame their interactions with their curriculum in the following
year.

Affective vs. Cognitive Variables in Teaching

Bonnie focused on affective rather than academic variables related to
teaching. According to Bonnie a good teacher was one that was responsive
to students' needs and interests. The teacher planned her instruction so
to "accommodate students both emotionally and intellectually."
A classroom led by a "good teacher" was engaging for all students
and involved them in collaborative problem solving.

This focus had served as a framework for her assessment of the programs
both at the beginning as she had joined the project and later as she implemented
the curricula in her classroom. Though she appreciated the fact that the
initial use of the programs had grabbed her students' interest as they had
eagerly engaged in mathematical activities, she attributed her hesitation
towards their use later in the year to her students' own lack of interest
in working on the new programs. Projecting that similar situation might
arise in the following year, she had concerns about creating a unified instruction
for all students.

They wanted structure-- They kept asking why they were doing these
things-- when they were going to do what they were supposed to do-- They
did not see what we were doing as doing mathematics-- This was discouraging
to a lot of them-- Some of them kept asking if they could choose something
else to do-- I felt my instruction was disjointed-- I was doing different
things with different kids and this was not good for them-- Some of them
felt that if they worked on the new curriculum it meant that they could
not do the regular math stuff and those that wanted more traditional stuff
where not working with others-- I know that this is going to happen again
this year-- I need to have lots of back-up plans.

In contrast, Gina's perceptions of good teaching were driven by her view
of "developing students' cognition," and "students learning
mathematics." Gina's references to the role of the teacher as one who
"helps students grow mathematically" consistently emerged as she
identified her own role within the learning environment as one who "posed
good problems," and "moved them along."

For Gina teaching mathematics consisted of "stimulating students'
thinking" and "waiting for students to react." For her, teaching
was a constant give and take between the teacher and the learners. For Gina
the most salient features of a good mathematics teacher included her knowledge
of mathematics and the learners. The teacher also knew how to use the instructional
tools and materials to evolve students' cognition.

It is really easy to say well, some kids get it and some kids don't
get it--So, just give them what they can do and let them work at their
own pace-- But, I think when a student is challenged, and when a student
really understands something then the talk about "lowering expectations"
is meaningless. I think we have a problem in middle school math teaching--
They keep talking about keeping them happy and all that but what it boils
down to is that when they are in my mathematics class, they are there to
learn mathematics-- My job is to make it accessible to them and to help
them see that they can do it-- Now, if I keep doing the old stuff that
never worked, it won't help much would it?

Role of the Curriculum

Each teachers' judgment of the programs was influenced by what seemed
to be most compatible with their images of teaching, and what they identified
as major dilemmas in teaching. The usefulness of a curriculum was determined
based on the extent to which it assisted them in resolving those dilemmas.
Moreover, their evaluation of curriculum programs depended upon how they
fitted in within their teaching structure as portrayed by their images of
ideal teacher and classroom. Therefore, the expectations of the curriculum
by teachers were of different nature. For Bonnie, the anticipation that
the curriculum would resolve the learning issues in class was prevalent.
Her perception of a good curriculum as one which "helped students understand
the mathematical concepts easier and better," dominated her judgment
of the programs. Her ideal curriculum was one that worked independent of
the teacher and accommodated all students' needs and their various learning
style.

Bonnie's lack of satisfaction with her old textbook was partially due
to the fact that it did not provide opportunities for students to "work
within meaningful contexts that facilitated their learning of mathematical
ideas." Not satisfied with the drill and practice routine of her old
textbooks, she appreciated the fact that the new curricula embodied contexts
that were removed from only paper and pencil sit work. Moreover, she valued
the fact that the new curricula provided her students a chance to "show
what they could do rather than what they could not do all the time."
However, when she discovered that her expectations of what it is like to
use a Standards-based curriculum conflicted with the realities of the classroom,
disillusionment naturally occurred (Cole & Knowles, 1993). The failure
of the Standards-based curriculum to overcome the problematic learning issues
was what she disliked most about their use. Bonnie's idealized image of
Standards-based textbooks was shattered when she discovered that the use
of the programs placed even greater demands on her as the teacher within
the learning environment.

I expected the curriculum to do a lot more for kids-- I expected
them to help them understand the concepts better so I wouldn't have to
go over the same things that I always did and answer the same questions
that I always did-- it just did not work that way-- There were a lot of
interesting things that my students did-- A lot of interesting activities
that got them going but at the end we were at the same spot that we were
before-- The only difference was that at least my students could do some
of the basic thing that they needed to do-- I had to push a lot of thing
at the end because I just waited and waited for them to learn how to do
certain thing-- Thinking that okay so these are supposed to help them do
it.

As opposed to Bonnie who viewed the role of the curriculum as one which
facilitated students' work, Gina perceived a good curriculum to be one which
made the "teacher's agenda more accessible to students." According
to her, a good curriculum was one that facilitated her work as the teacher.
For her, a curriculum program was valuable if it provided her with opportunities
to reach more students mathematically. For Gina, Curriculum program was
only a complementary component to a teacher's instruction. She repeatedly
articulated that she viewed the role of the curriculum as one that "supported
the teacher's instruction," rather than "dictating a particular
order to it."

A curriculum can never be perfect. After all, a textbook is written
by a group of people who think they know what should be done, and if we
are lucky they are written by some people who have a sensible way of thinking
(laughs)-- It is the teacher who has to use the curriculum and implement
it-- I still have my goals and my plans-- I look at a textbook and think
how can it make my job easier in accomplishing what I want to accomplish.

Mathematics Learning vs. Active Learning

As discussed earlier in previous section the problematic teaching issues
identified by teachers were of different nature, and related directly to
their particular focus in teaching, and those characteristics they attached
to productive instruction.

As it was most conducive for Gina to "teach mathematics," for
Bonnie the most important feature of a learning environment was "active
involvement" of all students in lessons. The differences in teachers'
views impacted how they perceived the programs initially and whether they
sustained momentum in using them throughout the year. This was explicitly
addressed by teachers several times during our interview, and confirmed
later in their practice as we observed their classrooms. Gina identified
the content of the programs as the major influence on her desire to continue
using them in her class.

I like the programs because they have such good maths in them--I
find it interesting when they say that there is no mathematics in the new
curricula. There is so much-- I like the fact that I don't have to keep
searching for good stuff all the time-- I have it all in the same unit--
I find myself asking a lot of what if questions, I have to, the program
is not doing it for me (Gina)

As Gina's analysis and assessment of the curricula was based on what
they had to offer her students mathematically, for Bonnie this analysis
was directly connected to the artificial consequences of their use in the
classroom. As she had initially appreciated the fact that the programs provided
opportunities for all her students to "do something in class,"
and to "read and write," a lack of direct connection between those
activities and her desired learning outcomes created dissatisfaction with
the programs on her part. As she continued using the programs, and as the
"newness of the activities faded", students' desire to work on
investigations had declined. The students' lack of interest and their constant
questioning about "why they had to do things that were not mathematics,"
declined Bonnie's own willingness to use the programs in her class.

You know 6 months into the year, and we had done all these group
activities and all they started telling me that they wanted to do other
stuff-- That they did not see why they had to do the programs because it
was just so much easier to do the worksheets. The only ones that really
showed any interest in using them were the ones that never did do so well
with worksheets-- They made some really neat projects, I had hanged them
all in the hall and I kept reminding them of all that they were learning,
and the problem solving stuff-- It really did work very well at the beginning,
but half way into the year, it just wasn't there anymore-- I was back to
where we were just as the year had started.

Teachers' Classroom Climate and Structure: Site Visits

The teachers' daily activities and classroom organization had many similar
features. In both classrooms, majority of the class time was spent on small
group work, and students-lead activities. In both classrooms students appeared
comfortable asking questions, working in small groups, sharing ideas with
others in large groups, and interacting with the teacher. Moreover, in both
classrooms majority of the students tended to stay on task throughout the
class session. In spite these similarities, different instructional patterns
also existed. These differences related to the degree of "deliberative
practice" that was exhibited during their sessions. As Gina seemed
to have very specific goals in what she did, including the type of problems
she posed in class, or the frequency of the use of whole group discussions
she conducted, Bonnie seemed less concerned about her own role and the impact
of her instructional decisions on students' mathematical learning. Gina
spend a greater amount of ankles time to whole group discussions of ideas.
She assured to ask students, both at the beginning and at the end of the
session, synthesize their findings in large group, and to compare solution
methods. She also consistently posed extensions of activities done in class
to further expand students' thinking about the concepts and their problem
solving activities. Large group discussions about students' own investigations
with the exception of one or two isolated occasions were non-existent in
Bonnie's classroom. Moreover, Bonnie's focus tended to be more on "covering"
the units and extensions of activities were not shared with students.

It also became evident that in spite the fact that teachers were expected
to use the new textbook at all times, the frequency of the use of materials
again varied from one teacher to another. Most frequently, Bonnie supplemented
the new materials with the textbook she had used in previous year. Though
she used the students' units and workbooks for ankles activities, she reverted
back to the old textbook in attempt to structure her instruction within
the bounds of the textbook she had used the year before. Though Gina began
the year using supplementary materials to the textbook, her use of these
materials declined over time.

Although at the first glance the most visible difference between the
two teachers appeared to occur within the realm of their general pedagogical
preferences and how they structured their class time, more settled differences
between their practices emerged gradually and as we continued to observe
their instruction. These differences for the most part appeared to be a
function of teachers' mathematics knowledge. In a general sense these differences
were revealed in every aspect of their teaching including their planning
procedures, their curricular goal setting, their method of handling students'
questions in class, and ways in which they dealt with students' thinking
as they encountered different concepts and algorithms. In addition, we came
to recognize the impact of teacher's content knowledge on how they structured
their class time. The collective impact of these elements determined how
each teacher reflected on her practice, interacted with students, and ultimately,
judged the curriculum.

Mathematical Understanding

It seemed that for both teachers the ability to recognize familiar mathematics,
or understand the mathematics presented in activities suggested by the program
was critical in whether they used the activities in class, or how they used
them with students. Moreover, this familiarity with, and understanding of,
mathematical ideas framed how they dealt with students' thinking and whether
they were able to connect their students' thinking to a mathematical structure.
This was fundamental in whether certain mathematical tasks were pursued
or abandoned by teachers. At the outset, this mathematical understanding
served as a basis for teachers' use of the programs on a consistent fashion.
A crucial factor in whether the materials were valued by teachers depended
heavily on their own content background and confidence in their knowledge
of mathematics.

Gina's approach to both mathematics teaching, and ways of using the textbook
was drastically different from Bonnie's. During the months that we observed
teachers, Bonnie frequently reported on not using certain sections of the
textbook, skipping certain activities and not spending too much time on
others. She also reported on replacing certain assignments and investigations
with more traditional materials she had used in the past. Gina also deviated
from the textbook often. However, this was more in the context of changing
the order and sequence of activities suggested by the textbook rather than
eliminating its parts entirely.

Gina's practice was framed by an orientation towards a problem based
instruction, and a substantially rich content knowledge base. These two
elements granted her the means to seek out every opportunity to make mathematical
ideas that were only implicitly addressed in the program explicit in her
instruction. Moreover, the fact that the curriculum complemented her approach
to teaching and facilitated her work to reach her content objectives, gravitated
her towards their use. It was clear that for Gina, drawing mathematical
connections from the activities was easier. Therefore, her appreciation
of the content of the program was more natural and thus, her desire for
the use of the materials more visible.

Though Bonnie demonstrated great patience towards students' who experienced
difficulty with the activities, and showed flexibility in how she worked
with different students as they engaged in individual and small group work,
this flexibility was not visible in how she discussed mathematics. Bonnie
avoided teaching those activities she did not feel comfortable with mathematically,
or those that she failed to view as mathematically significant. This, coupled
with the fact that students had begun asking "why" questions that
addressed the mathematics content of the activities they completed in class
to which Bonnie could not adequately respond, created an intense and even
less secure situation for her. It was evident that when Bonnie did not have
immediate answers to the students' questions, these questions were either
dismissed or not addressed by her.

Bonnie's lack of ability to deal with students' conceptions in non-routine
contexts created severe emotional set backs for her. At times she found
herself completely helpless in resolving the conflicting situations that
emerged as the result of students' sharing of ideas and of their thinking.
Her lack of ability to utilize prior knowledge or the work students had
done in previous sections to engage them in productive investigations, lead
to serious emotional disequilibrium on her part during her instruction which
perpetuated students' lack of interest in participating in class activities.
Moreover, these events motivated her need to revert back to an orientation
towards a more traditional instructional methods in which room for inquiry
and students' investigations were limited if not non-existent.

The following episode provides an example of this situation in one of
Bonnie's classes. The vignette comes from a session during which Bonnie
had just closed a discussion on the division of fractions. Students had
spent time using manipulative as they operated on various fractions for
three weeks. They had completed several investigations which implicitly
encompassed division of fractions without the use of algorithm. During the
first 10 minutes of class time, Bonnie presented the class a brief lecture
on how to divide fractions using the standard algorithm. She then provided
students with a handout which included 7 exercises involving division of
fractions. She asked students to form groups of two and to solve the exercises.
Two students argued that they did not understand why the algorithm worked
as it did and asked Bonnie for explanation.

S1: I don't get it-- Why do we change the division to times?

Bonnie: Anybody wants to explain why we do this?

S2: Because we flip the second fraction?

Bonnie: That is good-- Right, we change the division sign to multiplication
and then flip the second fraction. Like this: if we have 1/2 : 1/4, it
is like asking how many rods of length 1/4 units do we have in a length
of 1/2 unit? The answer is what? (pause) You can use your rods to do this--

Students use the cuisinaire rods to model the problem...

Bonnie: So, what is the answer here? Jim?

Jim: 2.

Bonnie: Good-- Now use your rods to do the problems I assigned to
you.

After a fifteen minute practice time, Bonnie asked students to come
to the board and to share their answers with others in class. As each student
shared specific answers to each exercise, Bonnie asked if others agreed
or disagreed with the response. John raised his hand and suggested a new
strategy.

Johnny: See, this is what I am doing-- If I have

I will do this:

I can do this for any fractions:.. See, like this one:

I just don't get the one you had-- Can I use this Ma. D.?

Sunny, Johnny's group partner argues that she does not believe Johnny's
algorithm is right since it does not seem to work for multiplication of
fractions. She, too, asks Bonnie to identify why she couldn't use a similar
procedure for finding the product of fractions.

Bonnie, looking at the procedure moves to the back of the class and
sits next to a student... The students are waiting for Bonnie to resolve
the situation...

Bonnie: We use the common denominator when we want to add two fractions...
For division or multiplication we really don't need to use the process
of finding the common denominator at all... It is all so easier.

John: But, Ma. D, it always works!

Bonnie: I wonder if we could find a situation when it doesn't work...
What do you think Johnny?

Bonnie asks students to look over the rest of the exercises and to
continue sharing their answer.

During the post session interview Bonnie expressed great discomfort with
the situation that had occurred in class and the fact that she could not
explain why "the algorithm was wrong."

I don't know why they don't get it... We have spent at least a month
working with C-rods and doing explorations-- I thought they would get the
flipping and multiplying of fractions right away-- It is the same problem
that my students had last year and even then we did use as much manipulatives...
The whole time we spent on this unit does not seem to be helping them much....

Rather than trying to make sense of the procedure, she immediately dismissed
its merit based on her own lack of familiarity with the relationship. Bonnie's
reluctance in dealing with the situation that had emerged, and her unwillingness
to think through the procedure that her student proposed in class was evidence
of her inability to connect students' thinking to a mathematical structure.

Though Bonnie avoided talking about what had happened in class, she chose
to emphasize her students' lack of preparation for understanding the basic
skills even in the presence of the Standards-based curriculum.

I should have introduced the algorithm earlier and given them more
practice. I guess the assumption that by following the guidelines they
would be able to do what they are supposed to be doing is wrong...

Bonnie neither attempted to understand why the procedure worked, nor
showed much interest in pursuing the topic with us further. Indeed, this
session served as a benchmark in her attempts at restructuring her classroom.
Following this session, Bonnie became more structured in her teaching. To
avoid unexpected discussions, she spent even less time on whole group discussions
and dealt with students mostly on an individual basis. Bonnie's hesitation
towards using the new standards-based textbook became quite visible at this
point and her reluctance to comply to the textbook's suggestions more apparent.

Teachers' Planning

The impact of teachers' own content knowledge came to play a significant
role in how they planned their instruction as well. A major difference between
the work of both teachers was the degree to which each one studied and assessed
the content of the program. This assessment was framed by both a mathematical
and pedagogical standpoint. Moreover, this assessment was a major influence
on each teacher's judgment of the program. This assessment served as a lens
through which the program was viewed by the teachers. Moreover, it shaped
each teacher's instruction on at least two levels. On the one hand, it provided
each the ability to draw important mathematics from the activities and share
them with students by synthesizing ideas in class from a global perspective
rather than individual pieces in isolation. On the other hand, it granted
them a sense of ownership of the materials to be taught in their classrooms.
Gina's extensive planning allowed her to feel successful in productively
implementing the program to accomplish her instructional objectives, whereas
a lack of extensive planning in Bonnie's case lead to dissatisfaction with
how the program was played out in class and perceived by students. Indeed,
the teachers' planning procedures matched nicely with their own specific
orientation towards teaching and their curricular expectations.

An important part of Gina's planning was her extensive review of the
materials she used in class. This included reviewing the entire unit rather
than individual lessons. Mathematics content and how and where related mathematical
ideas were presented in the program was important to Gina. She spent time
reading the units carefully, knew which ideas were visited throughout different
modules and investigations, and the sequence in which different concepts
and/or algorithms were developed. This understanding allowed her the opportunity
to act "flexibly" during her sessions, to move back and forth
between the activities, and as she posed questions to class. This ability
was strengthened by her own interest in seeking mathematical connections,
and mathematical problem solving.

Bonnie also studied and assessed the units she taught. However, her study
was mostly oriented around examining the Teacher's Guide and following its
suggestions for instruction. Her lessons came directly from the Teacher's
Guide as she followed the sequence from the book. For the most part she
did not include long range goals for the content development or learner
objectives beyond what was proposed by the program. This day to day approach
to planning lead to a number of problems that impacted her instruction on
both local and global levels. Absence of specific connections among activities
used in class and specific unit goals made it difficult for her to establish
and maintain a clear focus to guide instructional decisions. The purpose
of the activities she used in her class were often unclear to her and ultimately
unclear to students. She frequently presented a disjointed explanation of
topics, or discussion of problem situations assigned for students explorations,
and asked questions that were structurally peripheral to the intent of the
activity.

The differences between the teachers' level of assessment of the materials,
and their instructional priorities manifested themselves in their reflections
on their practice, and how they judged the curriculum program. Among the
8 classroom visits we made the one that exemplified these differences most
clearly came from a session during which teachers began their session as
they engaged students in a problem solving activity suggested as a warm
up exercise in the curriculum program. Both teachers intended to use the
activity in class. However, Bonnie had not realized the potential of the
problem for generating multiple solutions. We have included scripts from
our field notes because they vividly portray the teachers' approach to teaching
mathematics in a way that literal language fails to capture.

Problem 1:

It is Karen's birthday and Karen's mother bakes 13 cookies. People
at the party take turns picking cookies from the tray. If Karen takes the
first cookie and the last, how many people were at Karen's party?

Gina's Session:

Gina distributes the problem among students and ask everyone to work
on the problem first individually till they arrive at an answer and then
to get in groups of two and share their solutions with one another. She
asks each student to read the problem and make sure they understand the
problem first. One student raises her hand and asks if the each guest takes
a cookie. Gina invites others to comment on the question posed. "some
could take two and some may not want any cookie," the student comments.
Gina asks students to assume that everyone takes exactly one cookie at
each turn. As students begin working on the problem, Gina picks up a bucket
of Deine's unit blocks and places 13 blocks on each table. Without talking
to students she begins walking around the groups. Following a 5 minute
individual thinking time, Gina asks students to make sure they get into
their groups to share their final solutions. Students, as they are sitting
at their desks turn around and begin talking with one another. Another
10 minutes is passed before Gina calls everyone in the large group. Gina
asks for volunteers to share their solutions with others in class. One
student suggests that she thinks the answer is 12. Gina asks if others
agree or disagree with the student. Five students raise their hands in
agreement. One student asks if she could see the solution method. Gina
asks S1 to come to the board and show his work. The student asks if she
could use the overhead project and his blocks. She tosses 13 blocks on
the overhead projector, removes the first and last blocks, explains that
those two blocks represented the cookies Karen had taken and claims that
other guests each must each have taken one of the remaining cookies. Gina
looks silently at the class, waits for thirty seconds and asks if everyone
is clear with the explanation. All students nod their heads in agreement.
Gina asks if there are other answers to the problem. One student asks if
the problem meant that Karen had taken only two cookies and whether it
was possible for Karen to have more. Gina smiles and asks others if the
problem stated that Karen had only two cookies. Majority of the students
agree that it did not. Gina prompts again if in light of that information,
others had other solutions. The student who had asked the question suggests
that his answer is 2 and asks if she could show his way of thinking to
others. Gina invites the child to come to the board. Using the previous
student's model, she counts the 13 cookies to show that with two people
at the party, Karen would get both the first and last cookies. Gina asks
others if they thought the solution was reasonable and if there were other
solutions. No-one comments. Gina asks students if they could use S1, and
S2's model to see whether other combinations of numbers would be possible.
A minute later, one student suggests that she could agree that 3 guests
would also work. Another student asks is it is possible to have 4. Gina
asks them to use the blocks to test the question. She asks them to work
in groups of two and to determine all possibilities. Following another
5 minute discussion, students offer that numbers 2,3,4, 6, and 12 would
all work. Gina looks pleased. She is in front of the classroom. She is
silent for a minute and asks students if they could explain why all these
numbers satisfied the initial condition. There is no answer from the class.
Gina asks if they recognize any relationships between the numbers they
identified and 12. One student suggests that all numbers are even except
for 3. Gina asks them if they recognize any other relationships. Students
remain silent-- Gina asks if they remembered what they had talked about
earlier last week about finding factors/divisors of different numbers and
whether they saw any connections between what they had done there and this
particular activity.

[Field notes, December 14, 1996]

During the post-session interview, Gina suggested that as she had reviewed
the warm-up activity the night before, she had recognized the question as
a good "problem solving" situation that reinforced her instruction
on divisors of integers a week earlier. Gina, stating that she had not noticed
the significance of the problem at the first glance herself, admitted that
she had almost thought the question had only one solution. However, once
recognized the mathematics of the problem, Gina "knew that it was going
to be more than just a warm up problem."

I knew that this was going to get them going. This is a really rich
problem. And, its context is so familiar that I knew they would not hesitate
to attack it. It was worth my entire class time. I wished I had asked them
a couple of other questions when they came up with the generalization--
When I taught the topic of factors and divisors of different numbers a
few of students had difficulty seeing that when they say for example 33=3
X 11 both 3 and 11 were the divisors of 33. They would say 11 or 3 but
frequently not both-- This helps them see that. I would also have liked
to ask if they could come up with some extensions of the problem themselves.

Gina's assessment of her session was more intellectually than emotionally
based. Though she was pleased with the fact that all students were engaged
and involved in the activity, she felt unsure about when to intervene and
to what extent help children synthesize the content of the problem.

I had a feeling that I should have asked a couple of more what-if
questions at the end but I was not sure whether it would kill the discussion
at that point or not. I did not want to push them to realizing that final
connection, I wanted them to do it on their own-- I should have asked an
extension question at that point-- Like the one you suggested, using 25
with all other conditions staying the same- You know with 24 I could have
helped them made the connection even faster. At that point I just did not
think of it. I will do it with the next group.

In spite the fact that Gina was not pleased with how she had ended the
session mathematically, she felt more prepared for her next class session.
she modified her plan to take into account the new ideas for further extending
the problem if students failed to make the mathematical connection she desired.
Getting a better sense of how students would react to the problem she was
determined to spend more time during the second session on synthesizing
the content along with students. Connecting the content of the activity
to the overall value of the new program, Gina indicated that what she most
valued about the new Standards-based curriculum was the richness of the
activities which allowed students to explore ideas from a perspective that
was application oriented and meaningful.

As I look at the entire program, I notice that most activities are
equally as rich as the one I did in class today. It is making my job so
much easier-- Before I had to constantly be on the look for good materials,
now I have it all at one place and if I feel that I need to supplement
them with other activities to make sure the concept is established for
all of them, I still can do that. These are really good-- I mean really
good mathematically. I am still trying to get a hang of what comes where
and when but I know once I go through it once this year, it will be even
better next year.

Bonnie's Session

Bonnie had intended to use the "cookie problem" as a part of
her warm up segment and was prepared to begin a unit on fractions for the
day. The way events unfolded during her session, however, not only forced
a change of plans on her part, but also created a situation in which she
became uncomfortable with the mathematics discussion she had to lead in
class.

Bonnie assigns the warm up problems. Following an 8 minute work time,
Bonnie asks students if they are ready to present their solution to the
group. Several students raise their hands. S1 explains that his answer
is 12-- it seems that the answer is consistent with what Bonnie considers
as the correct answer. "Good work S1. Does everyone agree with the
solution?" Bonnie asks. S2 suggests that her answer is 2 and not 12.
Bonnie looks at the problem again-- She asks the student to go to the board
and to explain her strategy. S2 draws a circle and places 2 lines around
the circle... she counts starting from the line she identifies as Karen
up to 12 going around the circle 6 times -- Bonnie looks surprised. She
is looking at me for assurance. "What do you think class? Which do
you think is right?" S3 suggests that his answer is different from
the others and that she thinks there are 3 people at the party. Students
are quiet-- Bonnie asks S3 to show his work-- S3 using the same diagram
S2 has used, suggests adding another line around the circle representing
another guest and shows by the means of counting and going around the circle
4 times why she thinks the correct answer is 3. "Do you see what I
have done?" she asks Bonnie-- Bonnie nods her head in approval-- S4
goes back to his chair-- Bonnie looks puzzled and continues to read the
problem . A few seconds of silence passes as students look at Bonnie to
tell them what to do-- In the meantime, S4 asks if his answer, 6, could
be right- He, too, is asked to go to the board and to show his solution--
She specifically asks students to pay attention to S4's argument-- S4 extends
other students arguments by adding 2 more lines around the circle. In the
meantime S5 and S6 simultaneously ask Bonnie why their answer of 12 could
not be correct. Long pause by Bonnie as she looks at the solutions offered
by the group. "I think this was a really good problem. We all have
different answers and they all sound right... (pause) This is real problem
solving, you see! You all did very well class." S3 asks which answer
she should record on her paper-- "I don't understand why these answers
all can be right-- I still think mine is right! Should I write my answer
or everybody else's?" "Just put down all the answers so you could
think about it some more," Bonnie responds. All students begin recording
the answers on their papers.

[Field notes, December 14, 1996]

In the post-observation interview, Bonnie suggested that she was happy
with the outcome of her session. She felt that she had managed to engage
nearly all students in discussion and that she had "prompted students
to share their thinking." Although she had concerns about not knowing
what the answer to the problem was, she felt good about the fact that in
spite her own insecurity she allowed students to present their solution
methods to group. "The important point is for them to see with problem
solving there can be a number of different answers." In spite the fact
that the discussion took the entire class time (rather than the 10 minutes
she had anticipated) Bonnie was also pleased that she had allowed students
to work at their own pace and without placing time constraints on their
activity. She felt this element of "flexibility" was crucial for
children's sharing of ideas and for their learning. She felt that she facilitated
students' discussions even though she did not really know what the problem
was all about when she casually assigned it to groups.

Although Bonnie's own evaluation of the session was positive as she felt
validating all students' feelings and ideas were important, her realization
of her own lack of understating of the mathematics involved in the activity
was disheartening to her. This created a feeling of inadequacy on her part
that prevented her from using the activity again in her other classes.

You know I really felt awkward there. I just did not know what was
happening for a while. I guess I just couldn't think right on the spot.
I am still curious why the answers are right... I am not going to do this
with my next class though-- It takes too much of the class time and I think
I should get through my lesson .

Although Bonnie valued problem solving as a way to promote students'
involvement, she revealed that at times she had difficulty sustaining that
spirit in class. In her attempt to maintain a flexible approach, she failed
to provide opportunities for students to contribute to the lesson even when
she initially requested their input, or to use their contributions to make
a mathematical point explicit. This lead to not only student confusion and
helplessness in understanding mathematics, but also her own dissatisfaction
with the progression of lessons in class.

Students' Thinking

A strong manifestation of teachers' substance of mathematical knowledge
revealed in how they dealt with students' thinking during their sessions.
In fact, an important influence on teachers' judgment of the program was
whether they recognized children's patterns of mathematical thinking as
they engaged in solving problems, and their skills in justifying those thoughts
mathematically.

For Gina, working with children's thinking and conceptualizations as
articulated during problem solving episodes and group discussions served
as an integral component of her practice. She relied heavily on her students'
input and insights for further expanding her instruction. She comfortably
moved away from textbook assignments to discuss a particular point that
had derived from students' discussions and their solution methods. This
flexibility created a medium in which she addressed critical mathematical
points in contexts that were fresh and meaningful to students. As she planned
her lessons, she took into account cognitive obstacles students could potentially
experience as they encountered certain topics and assured to prepare additional
questions and activities that facilitated students' work as they thought
about those topics. She allowed students to explore ideas first, and later
by posing related questions helped them consider alternative explanations
and procedures. In this way, she confronted students' under- and over-generalizations,
or misconceptions in a constructive manner.

Gina's comfort with mathematics allowed her to maintain an open forum
in her classroom. She encouraged students' questioning, felt comfortable
working with student generated solutions in order to guide their activity
towards a desired piece of mathematics. Gina demonstrated great flexibility
in modifying her lessons based on the observations of the students. To her,
new insights and approaches to problems were avenues for further problem
solving on her part. Understanding students' thinking was "an exciting
part of teaching" that she eagerly embraced. She was willing to "pause"
during her instruction, try to make sense of what students had proposed
along with others in class, and to find a mathematical explanation that
justified their thinking. Indeed, her awareness of the multiple solutions
and thinking strategies motivated her to share those ideas with her students
in order to provide them with alternative methods of thinking about concepts,
and ultimately, "to do mathematics in greater depth."

The one vignette that is most illustrative of the profound impact of
Gina's content knowledge on her dealings with students' work and thinking
comes from a session in which students had begun working on the topic of
geometric similarity. Prior to this session students had spent nearly a
week on several investigations that introduced them to formal definition
of congruent figures. The concept of similarity of shapes had not been introduced
explicitly in the textbook. Though children were assigned to review a set
of figures and asked to determine which they thought were similar. Following
this activity, students were asked to articulate what they thought it meant
to say two figures were similar. As students were assigned to work on these
questions, Gina walked around the different groups and reviewed their responses
to the questions. A number of students had suggested that squares and rectangles
were similar, and that if two pictures looked alike they could be considered
as similar. Naturally, students' responses did not include detailed attention
to accurate measurements and to proportionality of the sizes of the similar
shapes. Considering students' raw generalizations Gina decided to stop the
small group activities and began a whole group discussion on the concept
of similarity. The following is a verbatim transcription of the conversations
that took place in her classroom during that discussion.

Gina: What is the best way to create two congruent figures?

Many students raise their hands. Gina asks one student to respond.

T: I can trace the figure.

J: Ya, like put a piece of paper on top and just copy it... we know
that they are going to be the same thing.

Gina: Let's do that together and see what we get-- She places a transparency
slide on the overhead projector and draws a rectangle. Placing another
transparency on top of it she traces the rectangle with a different color
marker.

Gina: Okay-- Let's see-- what do you notice about the two shapes
that we have here? (pause) How are they similar...

K: They look exactly the same--

Gina: What else? Tony?

Tony: I don't know-- they are just the same-- Like they are just
copies of each other.

Gina: Let's go beyond that and try to describe what we see in terms
of what we have talked about in the past-- Like, what can we say about
their angles? What can we say about their perimeters? What can we say about
their areas?

Students shout out that they are the same.

Gina: Right you guys-- This is really cool-- Their areas are the
same, their perimeters are the same, their angles are the same, and the
length of their sides are the same-- have I missed anything Jacquie?

J: Nope--

Gina: Good-- Now I want you to think about situations when we have
two things that look alike but are not exactly the same size-- Let's forget
the examples in the book and think of real life situations-- What do you
think? (pause)

S: Like in the movies?

L: Oh, ya-- in the pictures too.

Other students nod their heads in agreements.

Gina: Cool! So, when we go to the movies what we see on the screen
is actually similar to those images that we have on those little films...
Why do you say they are similar?

H: They are the same people only bigger--

Gina: tell us more Hillary--

H: Well, it is like Tom Cruise but a whole lot bigger-- His face
is bigger, but the same thing-- His body is the same thing, only bigger--

Gina: How about details in his clothing, his face?

T: Well, it has to be the same only bigger--

Gina: Right-- All the detail is there only bigger-- But could we
enlarge his legs say 10 times and his head, I don't know, say only 5 times?

Silence--

Gina: What would be wrong with that?

Jenny: It won't be right-- I mean, he won't look right-- Not as good
looking as Tom Cruise (laughs)

Gina: What else? Who else has something to say about this?

K: I think that it won't look right because if we make his legs bigger
say ten times then we need to make his head also ten times-- I guess that
is what it is-- If we don't enlarge it the same way for all parts it is
not gonna look right-- It won't look like him anymore--

Gina: What do you think Sam?

S: Hw would sure look funny- (laughs) I think some parts of him will
be similar but not the whole thing--

Gina: How can we make sure that what when we have two shapes, they
are similar?

P: I guess we can check to see if it is bigger or smaller the same
amount on all parts..

Gina: Good idea-- Let's see if we can do what P. said--

She turns on the overhead projector and projects the rectangle she
had drawn on the transparency on the screen-

Gina: Do You think these two shapes are similar (pointing at the
one of the screen and the rectangle on the transparency?

D: I think so--

Gina: How can we be sure?

D: It is only the same thing only bigger?

Gina: How much bigger do you think the one on the overhead projector
is?

Students decide that they want to measure each rectangle first. Gina
asks them to do so and also to look at the perimeters and areas of the
two figures and to decide whether there is a relationship between the two
sets of data. She also encourages students to move the overhead projector
and consider other cases.

Bonnie's instruction was also influenced by her students' thinking, and
her knowledge about children's common problems as they encountered topics.
Though her methods of dealing with those understandings were more directive.
She announced at the beginning what she knew of the potential problems,
and clarified the points prior to assigning activities. For the most part,
her method of dealing with the students' lack of understanding was doing
more of similar examples. Though, she listened to students carefully, and
encouraged them to share the process through which they had arrived at certain
answers she was not always able to make sense of those explanations or to
use them as contexts for further elaborate on ideas. Often, she failed to
recognize fruitful questions posed by students. At times, she pursued questions
and responses that lead to dead ends. Vignettes provided in previous sections
provide the readers with ample examples of these elements.

Bonnie experienced difficulty dealing with unexpected situations that
involved "flexibility" in working with the emerging discussions.
She was most fragile and felt inadequate when she "did not" have
an explanation for the student generated questions. This build up of sense
of inadequacy presented itself in how she structured her lessons. Her organization
of her lesson, and class structure was such that it avoided whole group
discussions for the most part. She strategically planned the time so there
would be more of one-on-one work with individual students, or small groups
rather than discussions that placed her in front of the classroom. In places
where she felt comfortable with the content, and the questions that emerged
she was willing to open the question to the whole group and served as a
moderator as students shared their ideas with others in class. Other times,
she presented procedures as if they were self evident and students should
accept and remember them without much thought. Such unbalanced instructional
practice created a personal turmoil for Bonnie. As she felt a lack of control
on what happened in class, she began to hold the curriculum primarily responsible
for the dilemmas she was facing.

Gina, on the other hand, seemed more than willing to invite students
to participate in open discussions of topics. She provoked conflict, and
for the most part tried to hold students accountable for resolving ambiguities
that emerged as they worked on different problems. However, she did regularly
intervene and moved the discussion in a direction that she felt appropriate.
She was enthusiastic to "lead students' thinking and talking into the
right channel." For Gina, the use of the programs was energizing. In
a general sense, it allowed her to be a part of the problem solving activities
that took place in her classroom. On several occasions she expressed great
satisfaction with the fact that she did not have to follow the "boring
structure" of the old textbook. The excitement generated as the result
of the use of the new curriculum motivated her to the point that she consistently
investigated extensions of the activities. She willingly shared those extensions
with her students and encouraged them to "think as mathematicians did."
This spontaneity in teaching, increased over time. Though using the activities
in the units she reorganized the presentation of the materials. In this
way, she followed the suggested structure in the teacher's guide only occasionally
and took control of how the curriculum was used in her class.

A Discussion

The data presented in this paper points at the impact of teachers' mathematical
knowledge, their image of teaching and of curriculum in light of that knowledge,
and on how a Standards-based curriculum was assessed and used in their classrooms.
Though it appeared at first that the amount of time spent on planning lessons
was a critical factor in how the program was used by teachers and whether
it was used in classes at all, it became even more transparent that the
teachers' knowledge of mathematics, their ability and skills in directing
mathematical inquiry among students in ways that were conceptual and on
a meaningful mathematical continuum was a crucial basis for their assessment
of the Standards-based curriculum and their implementation of it.

In order for teachers to help students grow mathematically, as it was
envisioned by the new curriculum, they needed to have an understanding of
the central ideas of mathematics, see relationships among concepts, be able
to reason mathematically, and have an elaborated knowledge base that included
multiple representation of concepts. The teachers needed to know how and
in what ways mathematical ideas were connected and to grasp what was structurally
central and what was not in discussing topics presented in the units. The
structure of the materials was such that the content discussion was not
around specific topics but oriented around open ended mathematical problems
some of which only implicitly addressed certain mathematical concepts. Therefore,
the teachers needed to know how to build on students' insights towards different
mathematical investigations and be able to connect those personal insights
to a specific mathematical idea. This, however, was not the case for one
of the teachers participating in this study. In fact, the absence of this
type of mathematical knowledge was a catalyst in one teacher's reluctance
towards the use of the Standards-based program, whereas its presence was
the underlying motive for the other teacher's enthusiasm towards using the
new curriculum.

Gina investigated each activity first hand with an attempt to evaluate
the mathematical connections embedded throughout the entire curriculum.
Her genuine desire and interest in "doing mathematics," and her
skills in mathematical problem solving allowed her to draw pedagogically
and mathematically important decisions throughout her instruction. These
decisions concerned both the curriculum she used, and how she chose to teach
that curriculum. The new program allowed Gina the flexibility she needed
to create a problem based instruction that was visibly shaped by her specific
mathematical goals.

For Gina, the use of the program was a challenge she took on eagerly.
She found the materials intellectually stimulating and mathematically significant.
It was evident that she spent a greater amount of time reading the activities,
familiarized herself with the ideas presented in the entire unit rather
than teaching one lesson at a time. The program suited her predominant teaching
orientation, and the problem based classroom routine she intended to establish.
She found the program useful in offering her students engaging activities,
and herself a chance to help students make mathematical connections. Many
of these characteristics were not present in Bonnie's practice.

Though Bonnie appreciated the fact that the programs provided students
engaging activities, she frequently was not successful in attaching mathematical
value to them. In addition, she failed to recognize how the program addressed
traditional content objectives. The link between specific concepts embedded
within the interrelated investigations were not obvious to her. For Bonnie
the use of the program and how she implemented it was closely connected
to her understanding of the content embedded in the activities. Her judgment
of the usefulness of the materials was a function of her ability to "realize"
the mathematical themes that could potentially derive from particular activities
and/or units. In cases where mathematical connections were not visible,
she was reluctant to use the materials in class. Absence of specific connections
among activities used in class and specific content goals made it difficult
for her to establish and maintain a clear focus to guide instructional decisions.
Moreover, absence of specific mathematical understandings made it even more
difficult for her to maintain a consistent mathematical structure in class.
For these reasons, there was a strong affective component attached to the
use of the Standards-based program by Bonnie. This affective component determined
how she conducted her lessons, interacted with students, and evaluated the
outcomes of her instruction on learners. She struggled constantly to monitor
the student input in order to address specific content objectives. Any deviations
from her already designed guideline created a feeling of insecurity for
her. This lack of security manifested itself in many forms. She changed
the classroom discussion immediately, focused students on specific pieces
she intended to cover, felt most secure when working individually with each
student and as the problematic teaching/learning issues were familiar to
her. Her use of the curriculum was a function of her confidence in drawing
mathematical connections herself. When realizing her inability to communicate
or direct relevant mathematical discussions in class she became reluctant
to use the materials and frequently refused to use them in her instruction.

Epilogue

The topic of what mathematics teachers need to know for productive practice
has long being debated. Although knowledge of the subject matter is probably
the most self evident kind of knowledge needed to teach, the amount of subject
matter knowledge needed to help children is a contested issue (Kenny 1998).
Attention in recent years has shifted from defining teachers' mathematics
knowledge in terms of the volume of knowledge, to the substance and character
of that knowledge. In recent years a number of studies have suggested that
teachers' background knowledge in subject matter affects how they represent
the nature of knowing to their students (Shulman & Grossman 1988; Ball
1990, 1988a,b, Shulman 1986; Carlson 1988; Enderson 1995; Manouchehri &
Enderson in press; Thompson et al, 1994). Although the findings of research
on the effect of teachers' subject matter knowledge on student learning
has been inconclusive, research acknowledges the need for teachers to develop
new understanding of the subject for themselves before they can help their
students engage in conceptual change in mathematical thinking (Lampert 1986).

Our research succeeded to provide some tangible evidence of how the content
knowledge of teachers come to play a role in their instruction, interactions
with students, and their curriculum decision making. In the absence of an
understanding of mathematics and knowledge about mathematical connections
on the part of the teacher participants in this study even sincere attempts
at creating an inquiry based instruction were futile. Moreover, this study
provided ample evidence that teachers' own mathematical knowledge shaped
their expectations of teaching and learning and influenced not only how
they interpreted the Standards-based curriculum but also how they used it
in their classes.

In the presence of a curriculum that builds heavily on student centered
learning and non-routine tasks, the mathematics teacher needs to have a
type of mathematical knowledge base that is detailed, allows for sound reasoning,
and mathematical problem solving. The substance of this knowledge is remarkably
different from recitational knowledge (Kenny 1998). This knowledge base
directly impacts the teacher's ability in advancing students' mathematical
thinking by assisting them connect their findings to a relevant mathematical
structure. In the absence of such knowledge the teacher faces many dilemmas
of both personal and professional nature. The interplay of these personal
and professional dilemmas create disabling environments for both the teacher
and the learners.

Our research also highlights the notion that in the presence of Standards-based
curricula the teacher needs to learn and obtain new knowledge about her
role in the classroom, re-examine her curriculum and instructional goals
in light of the new needs, and re-conceptualize the nature of her interactions
with the learners. We propose, without the loss of generality, that for
successful implementation of the Standards-based curricula the teacher needs
to have a deep understanding of the mathematical ideas and of the connections
among different concepts; a sound knowledge of children's cognition and
how they learn mathematics; and the "how to knowledge" necessary
for operating "flexibly" and "productively" within such
student centered setting. This how to knowledge is multi-dimensional and
includes elements related to methods of guiding students' inquiry, mapping
gradual development of both the content and learner's thinking, creating
a balance between fostering students' conceptual understanding while assisting
them acquire basic skills. For many teachers these require adopting a new
paradigm towards teaching, a re=conceptualization of their own role in the
classroom, and a reexamination of their own understanding of mathematics.
These, inevitably, require the teachers to be willing to enter a zone of
practice that is ambiguous and uncertain. For in reality while current mathematics
reform has inspired the development of many innovative mathematics curriculum
programs, these materials do not provide teachers with enactment strategies
that are necessary for their successful implementation. Indeed, our research
provided some evidence that the presence of these enactment skills, or their
lack of, serve as crucial constituent in whether teachers sustain commitment
to using the reform-based programs in their classes or abandoning them.

The identity of one of the teacher participants in this study was completely
shattered by her inability to construct practice that reflected the beliefs
that accompanied her ideal classroom. She wanted to have students to act
as mathematicians, engage them in learning but she did not know how to achieve
this goal. Her limited knowledge of mathematics served as an obstacle in
her ability to create the classroom environment she desired to create. She
knew that she wanted to create an open forum in which students engaged in
sharing of ideas, and social construction of knowledge but she did not know
how to implement, and sustain this conceptualization of mathematics teaching.
Her limited understanding of mathematical ideas, and lack of experience
in mathematical problem solving created the greatest obstacle in the instructional
routine she intended to establish in her classroom. In spite the fact that
she initiated mathematical dialogues in her class, she was not able to nourish
that discourse or help students connect their "conversations"
to a mathematical structure. This was devastating to her. Consequently,
she reverted back to teaching methods that were more structured and traditional.
Moreover, she became less willing to use the type of curriculum that oriented
around students' open investigations.

Recently Wilson (1998) suggested "Meaningful reform and genuine
change will result only if it is manifested within mathematics classrooms.
The role of the teacher is essential" (p.2). The findings of this study
substantiate his assertion as they stress that teachers need concrete images
that depict what it is like to teach in ways that are in accordance with
the reformed visions of teaching. In the absence of these images efforts
towards implementing reform are futile. Analogous experiences can provide
opportunities for teachers to experience the principles and concepts inherent
in a Standards-based approach to mathematics instruction. These authentic
experiences should explicitly address a re-examination of the structure
of the discipline, engage teachers in conceptual explorations of mathematics,
and assist them construct a mathematical and pedagogical point of view that
would serve as a foundation for successful implementation of reformed-based
curriculum and instruction.

Cooney, T. J. (1994). Teacher education as an exercise in adaptation.
In D. B. Aichele and A. F. Coxford (Eds.), Professional development for
teachers of mathematics, 1994 Yearbook (pp. 9-22). Reston, VA: National
Council of Teachers of Mathematics.